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Theorem tsrlin 14344
Description: A toset is a linear order. (Contributed by Mario Carneiro, 9-Sep-2015.)
Hypothesis
Ref Expression
istsr.1  |-  X  =  dom  R
Assertion
Ref Expression
tsrlin  |-  ( ( R  e.  TosetRel  /\  A  e.  X  /\  B  e.  X )  ->  ( A R B  \/  B R A ) )

Proof of Theorem tsrlin
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 istsr.1 . . . . 5  |-  X  =  dom  R
21istsr2 14343 . . . 4  |-  ( R  e.  TosetRel 
<->  ( R  e.  PosetRel  /\  A. x  e.  X  A. y  e.  X  (
x R y  \/  y R x ) ) )
32simprbi 450 . . 3  |-  ( R  e.  TosetRel  ->  A. x  e.  X  A. y  e.  X  ( x R y  \/  y R x ) )
4 breq1 4042 . . . . 5  |-  ( x  =  A  ->  (
x R y  <->  A R
y ) )
5 breq2 4043 . . . . 5  |-  ( x  =  A  ->  (
y R x  <->  y R A ) )
64, 5orbi12d 690 . . . 4  |-  ( x  =  A  ->  (
( x R y  \/  y R x )  <->  ( A R y  \/  y R A ) ) )
7 breq2 4043 . . . . 5  |-  ( y  =  B  ->  ( A R y  <->  A R B ) )
8 breq1 4042 . . . . 5  |-  ( y  =  B  ->  (
y R A  <->  B R A ) )
97, 8orbi12d 690 . . . 4  |-  ( y  =  B  ->  (
( A R y  \/  y R A )  <->  ( A R B  \/  B R A ) ) )
106, 9rspc2v 2903 . . 3  |-  ( ( A  e.  X  /\  B  e.  X )  ->  ( A. x  e.  X  A. y  e.  X  ( x R y  \/  y R x )  ->  ( A R B  \/  B R A ) ) )
113, 10syl5com 26 . 2  |-  ( R  e.  TosetRel  ->  ( ( A  e.  X  /\  B  e.  X )  ->  ( A R B  \/  B R A ) ) )
12113impib 1149 1  |-  ( ( R  e.  TosetRel  /\  A  e.  X  /\  B  e.  X )  ->  ( A R B  \/  B R A ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    \/ wo 357    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696   A.wral 2556   class class class wbr 4039   dom cdm 4705   PosetRelcps 14317    TosetRel ctsr 14318
This theorem is referenced by:  tsrlemax  14345  ordtrest2lem  16949  ordthauslem  17127  ordthaus  17128
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-br 4040  df-opab 4094  df-xp 4711  df-rel 4712  df-cnv 4713  df-dm 4715  df-tsr 14323
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